[Paper Review] Weighted trace cochains; a geometric setup for anomalies

Sylvie Paycha|ArXiv.org|Mar 14, 2005

Advanced Operator Algebra Research8 references3 citations

TL;DR

This paper introduces weighted trace cochains as a geometric framework to analyze algebraic and geometric anomalies in pseudodifferential operators via regularized traces. By extending the canonical trace to multilinear cochains weighted by a positive-order elliptic operator Q, it shows that discrepancies—arising from Hochschild coboundaries (algebraic) and covariant derivatives (geometric)—are finite linear combinations of Wodzicki residues, with explicit formulas derived via Mellin transforms and asymptotic expansions of heat kernels.

ABSTRACT

We extend formulae which measure discrepancies for regularized traces on classical pseudodifferential operators to regularized trace cochains, regularized traces corresponding to 0-regularized trace cochains. This extension from 0-cochains to $n$-cochains is appropriate to handle simultaneously algebraic and geometric discrepancies/anomalies. Algebraic anomalies are Hochschild coboundaries of regularized trace cochains on a fixed algebra of pseudodifferential operators weighted by a fixed classical pseudodifferential operator with positive order and positive scalar leading symbol. In contrast, geometric anomalies arise when considering families of pseudodifferential operators associated with a smooth fibration of manifolds. They correspond to covariant derivatives (and possibly their curvature) of smooth families of regularized trace cochains, the weight being here an elliptic operator valued form on the base manifold. Both types of discrepancies can be expressed as finite linear combinations of Wodzicki residues.We apply the formulae obtained in the family setting to build Chern-Weil type weighted trace cochains on one hand, and on the other hand, to show that choosing the curvature of a Bismut-Quillen type super connection as a weight, provides covariantly closed weighted trace cochains in which case the geometric discrepancies vanish.

Motivation & Objective

• To develop a unified geometric framework for analyzing anomalies in infinite-dimensional geometry and quantum field theory.
• To extend regularized trace functionals from 0-cochains (traces) to n-cochains (multilinear forms) using a weight operator Q.
• To distinguish and characterize algebraic anomalies (Hochschild coboundaries) from geometric anomalies (covariant derivatives of families of cochains).
• To show that both anomaly types are expressible as finite linear combinations of Wodzicki residues.
• To construct Chern-Weil type weighted trace cochains and identify conditions under which geometric anomalies vanish.

Proposed method

• Define weighted trace cochains χⁿ^Q as the finite part of the Mellin transform of the multilinear form t ↦ tr(A₀e⁻ᵗᵘ₀Q⋯Aₙe⁻ᵗᵘₙQ) on the simplex Δₙ.
• Use the Mellin transform of the heat trace to define meromorphic functions z ↦ χ̄ₙ,ₚ(z), whose residues yield Wodzicki residues of operator products.
• Derive an explicit formula for χⁿ^Q(A₀,…,Aₙ) as the weighted trace tr^Q(A₀⋯Aₙ) plus correction terms involving Wodzicki residues of Q⁻|k| and adjoint actions Aᵢ⁽ᵏⁱ⁾.
• Express the Hochschild coboundary bχ²ᵖ^Q as a finite sum of Wodzicki residues involving Q⁻|k|⁻¹ and higher-order adjoint derivatives.
• Apply the formalism to families of algebras over a smooth fibration, interpreting geometric anomalies as covariant derivatives of cochains with respect to a connection on the base manifold.
• Show that choosing the curvature of a Bismut-Quillen superconnection as the weight Q yields covariantly closed cochains, eliminating geometric anomalies.

Experimental results

Research questions

• RQ1 How can regularized traces on pseudodifferential operators be generalized from linear functionals to multilinear cochains in a way that captures both algebraic and geometric anomalies?
• RQ2 What is the precise structure of the discrepancy between the cyclic trace of a product and the weighted trace cochain, and how is it expressed in terms of local invariants?
• RQ3 How do algebraic anomalies—arising as Hochschild coboundaries—relate to Wodzicki residues in the presence of a weight operator Q?
• RQ4 In the context of smooth families of algebras, how can geometric anomalies be characterized as covariant derivatives of weighted trace cochains?
• RQ5 Under what conditions does the geometric anomaly vanish, and what role does the curvature of a superconnection play in this?

Key findings

• The weighted trace cochain χⁿ^Q(A₀,…,Aₙ) is given by tr^Q(A₀⋯Aₙ) plus a finite sum of Wodzicki residues involving Q⁻|k| and adjoint derivatives of the Aᵢ’s.
• The Hochschild coboundary of a 2p-cochain bχ²ᵖ^Q is a finite linear combination of Wodzicki residues with coefficients involving (−1)^|k| |k|! / (k+1)! and powers of Q.
• For p=0, the algebraic anomaly (b tr^Q)(A,B) is a sum over k of (−1)^k / (k+1) times the Wodzicki residue of AB⁽ᵏ⁺¹⁾Q⁻ᵏ⁻¹.
• Geometric anomalies in families are captured by the covariant derivative of the cochain, and vanish when the weight is the curvature of a Bismut-Quillen superconnection.
• The construction yields Chern-Weil type weighted trace cochains that are closed under the (b,B)-bicomplex when the weight is curvature of a superconnection.
• The formula for the Mellin transform of the multilinear form is χ̄ₙ,ₚ(z) ≃ Γ(z+n) TR(A₀⋯Aₙ Q⁻ⁿ⁻ᶻ) + ∑ terms involving higher-order adjoint actions and TR of Q⁻ⁿ⁻|k|⁻ᶻ.